Mizuho Enmansai

8/9/2019 Mizuho Enmansai

1/37Electronic copy available at: http://ssrn.com/abstract=1950001

An Analytical Hybrid Equity and Interest Rate Model

Shek–Keung Tony Wong∗

Abstract

In this article, we introduce a new pricing model for equity and interest rate hybrids that can capturethe effects of implied volatility skews and stochastic interest rates simultaneously. The proposed modelis a simple modification of an existing hybrid model often used by market practitioners. While both

models can be calibrated efficiently and accurately along the lines of Piterbarg (2005), it is shown thatthe proposed model is analytically solvable leading to an explicit equity price expression. Using thisexpression, we manage to establish an exact Monte Carlo simulation scheme for generic pricing purposes.In addition, it is shown that a pathwise approach along the lines of Glasserman (1996) can be appliedto the proposed model in a straightforward manner for the calculation of equity deltas and gammas. Forillustration purposes, we apply the results developed in this article to the pricing of conditional triggerswaps or Enmansai structures. Numerical results indicate that the exact simulation scheme developed inthis article is both efficient and robust. Moreover, both the proposed model and the existing model leadto similar prices, equity deltas, and equity gammas. However, owing to the exact simulation scheme, theproposed model compares favorably to the existing model due to its superior computational efficiency androbustness in pricing. As a benchmark, with a simulation size of 100K, calculation of price, equity deltas,and equity gammas of a 30–year Enmansai trade with semi–annual coupons typically takes less than 5seconds under the proposed model, while the existing model can take nearly 100 seconds to achieve acomparable accuracy.

1 Introduction

In practice, there exists financial instruments whose values cannot be fully characterized by liquidly traded marketinstruments. In such cases, it is typical that a model is needed to ’complete’ the price characterization. Equity andinterest rate (EIR) hybrids are particular examples of such instruments. The value of an EIR hybrid depend cruciallyon the stochasticity in both interest rate and equity price movements, yet cannot be uniquely represented by marketprices of liquidly traded interest rate and equity vanillas. In the context of financial modelling, this is equivalentto saying that the value of the instrument under a two–factor (2F) pricing model is significantly different from itsvalue under a one–factor (1F) pricing model. Consequently, a 2F pricing model is needed to capture the multiple risksembeded in the instrument and hence risk–manage these risks appropriately. On the other hand, for many existing EIRhybrids, the effects of implied volatility (IV) skews also play a crucial role in pricing, which adds another dimension

of difficulty in the pricing of such products. A detailed introduction of EIR hybrids and their risk characteristics canbe found in Overhaus (2006) for a detailed description.

The applications of 2F pricing models are hindered by their increasing computational intensity in pricing anddifficulty in calibration. While a simple 2F hybrid model with normal short rates and lognormal equity prices may besufficiently efficient in both calibration and pricing, it fails to produce IV skews commonly observed in equity optionmarkets. The challenge remains to identify a pricing model that can strike a good balance between computationalefficiency in pricing and calibration and the capability of capturing the essential risk factors associated with EIRhybrids. Piterbarg (2005) presents a candidate pricing model fitting these criteria. The Piterbarg model, tailoredfor the pricing of power reverse dual currency (PRDC) swaps, is formulated in a double–currency setting with theforex rate following a diffusion process with a CEV volatility function and the short rates in the two underlyingcurrencies driven by one–factor Hull–White (HW1F) models. Due to the analogy between forex rate modelling and

∗Global Methodology Office, Mizuho Securities, Tokyo, Japan. The contents of this article only reflect the views of the author and notnecessarily those of Mizuho Securities. The author is indebted to Tsukasa Yamashita, Hiroshi Nishida, Sebastien Gurrieri, and MasakiNakabayashi for useful comments and suggestions. All remaining mistakes are my own.

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2/37Electronic copy available at: http://ssrn.com/abstract=1950001

equity price modelling, this model can be easily adapted to the pricing of EIRs1. The popularity of the Piterbargmodel is particularly attributed to: (i) the existence of an efficient and accurate calibration procedure, and (ii) theease of control of IV skews. An existing model that is widely used by market practitioners for the pricing of EIRs isindeed closely related to the Piterbarg model. The existing model replaces the CEV volatility function in the Piterbargmodel with a displaced diffusion volatility function2. It turns out that calibration of this model can also be performed

efficiently and accurately along the lines of Piterbarg (2005).In this article, we present a new hybrid pricing model for EIRs that can incorporate the effects of implied volatility

skews and stochastic interest rates simultaneously. An efficient and accurate calibration procedure along the linesof Piterbarg (2005) is presented. Hinging on a simple modification of the volatility function of the existing modeldiscussed above, the proposed model can be solved analytically leading to an exact Monte Carlo simulation scheme forgeneric pricing purposes. The proposed model is also subject to some limitations that are similar to those discussedin Piterbarg (2005). In particular, it cannot be well fitted to convex implied volatility smiles. However, the currentarticle does not aim to provide a perfect solution to the pricing of EIRs, but rather a parsimonious pricing model whichis capable of capturing the crucial risk factors underlying EIRs and allows for fast and accurate model calibrationand pricing. The remainder of this article is organized as follows. The next section provides the model definition anda brief comparison between the proposed model and the existing model. In Section 3, we describe the calibrationprocedure while its derivation is given in the Appendix. In Section 4, we derive an explicit expression for the spotequity price and subsequently an exact Monte Carlo pricing framework for EIRs. An analytical formula for momentsof spot equity price is also given. In Section 5, we illustrate the results developed in this article through a few practicalpricing examples. Finally, Section 6 concludes the article.

2 The Model

2.1 The Short Rate Dynamics

Let r(t) denote the short rate. We assume that r(t) follows a HW1F process

r(t) = ψ(t) + y(t), (1)

dy(t) = −a(t)y(t)dt + σr(t)dW r(t), y(0) = 0, (2)

where W r(t) is a standard Brownian motion under the risk neutral measure Q. The parameters a(t) and σr(t) denotethe mean reversion speed and the short rate volatility, respectively.

In addition, ψ(t) is a deterministic function of time which can be used to achieve an exact fit to an initial yieldcurve. Let P (t, T ) denote the time t price of a zero coupon bond (ZCB) maturing at T . Under the HW1F short ratemodel, P (t, T ) is given by

P (t, T ) = exp (A(t, T ) − B(t, T )y(t)) , (3)

where

A(t, T ) = − T t

ψ(u)du + 1

2V I (t, T ), (4)

B(t, T ) = E (t) T t

1

E (u) du, (5)

E (t) = exp

t0

a(u)du

, (6)

V I (t, T ) =

T t

B2(u, T )σ2r(u)du. (7)

Assume an initial discount curve P = {P (0, t)}t≥0, where P (0, t) is differentiable in t. Then, the HW1F model can befitted exactly to P by setting

ψ(t) = −∂ ln P (0, t)∂t

+ 1

2

∂V I (0, t)

∂t . (8)

1

When pricing EIRs using the Piterbarg model, one may regard the forex rate as the underlying equity price while treating the foreignshort rate as the dividend yield.2Several studies reveal that CEV models can be well approximated by displaced diffusion models, see Marris (1999) and Svoboda-

Greenwood (2007) etc., for example.

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Following some simple arguments, one can solve (2) to obtain

y(T ) = 1

E (T )

T 0

E (u)σr(u)dW r(u), (9)

T

0

r(u)du =

−ln P (0, T ) +

V I (0, T )

2

+ B(0, T ) T

0

E (u)σr(u)dW r(u)

− T

0

B(0, u)E (u)σr(u)dW r(u).

(10)

For a detailed review of the HW1F model, see Brigo (2001), for example.

2.2 The Equity Price Dynamics

Let S (t) be the price of an underlying equity asset. In conjunction with the HW1F short rate dynamics, we proposethe following dynamics for S (t)

dS (t) = (r(t) − d(t)) S (t)dt + (β (t)S (t) + (1 − β (t))x(t)) σS (t)dW S (t), S (0) = S 0, (11)where d(t) denotes the dividend yield and W S (t) a standard Brownian motion under Q with

< dW r(t), dW S (t) > = ρdt. (12)

The variable x(t) is defined as

x(t) = S 0 exp

t0

(r(u) − d(u)) du

. (13)

Within the current model, the volatility parameter σ(t) influences mainly the level of at–the–money (ATM) impliedvolatilities, while β (t) controls the slope of implied volatility curves along the strike axis and is often referred to asthe skew parameter. Note that x(t) is lognormally distributed under the HW1F assumption for r(t).

As in many practical applications, we assume that the parameters β (t) and σ (t) are piecewise constants:

(β (t), σS (t)) =

(β 0, σ0), t ∈ [0, T 1),(β 1, σ1), t ∈ [T 1, T 2),...

...

(β N −1, σN −1), t ∈ [T N −1, T N ).

(14)

With this specification, one has the flexibility of fitting the model to a term structure of implied volatility skewsobserved in the markets.

2.3 Comparison to an Existing Hybrid Model

The proposed model is a simple modification of an existing hybrid model which is popular among market practitioners.Under the existing model, S (t) follows

dS (t) = (r(t) − d(t)) S (t)dt + (β (t)S (t) + (1 − β (t))F (0, t)) σS (t)dW S (t), (15)where F (0, t) denotes the initial equity forward price given by

F (0, t) = S (0)e−∫ t

0

d(u)du

P (0, t) . (16)

Hence, our proposed model can be obtained from the existing model simply by replacing the forward price F (0, t)appearing in (15) by x(t). Note also that the proposed model and the existing model coincide under the degeneratingcase σr(t) = 0, where both models reduce to a one–factor displaced diffusion (DD1F) model (e.g. see Rebonato(2004)).

Using (10), we may rewrite x(t) as follows

x(t) = F (0, t)exp

V I (0, t)

2 + B(0, t)

t0

E (u)σr(u)dW r(u) − t0

B(0, u)E (u)σr(u)dW r(u)

. (17)

Since the volatility of x(t) is generally much smaller than the volatility of S (t), we expect that the pricing behaviorsof the two models are similar in general. Note that the existing model is not solvable in general in that there exists

no explicit expression for the equity price S (t). This is indeed the main motivation for us to introduce the alternativeequity price dynamics (11). In a later section, we shall present an explicit expression of S (t) under the proposed modelwhich consequently leads to an efficient Monte Carlo simulation scheme for generic pricing purposes.

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3 Model Calibration

Calibration of the model involves the determination of the HW1F parameters, the correlation ρ, and the equity skewand volatility parameters. With the HW1F model, an initial yield curve can be fitted exactly using the time dependentparameter α(t) while the mean reversion speed a(t) and the volatility σr(t) are usually chosen to match market prices

of European swaptions and/or caps/floors. There exists a large body of literature on the calibration of the HW1Fmodel, e.g. see Brigo and Mercurio (2001) or Hagan (2000). The model correlation ρ, on the other hand, is typicallyestimated from time series of historical interest rates and equity prices. The main challenge lies in the calibration of theequity skew and volatility parameters {(β i, σi)}N −1i=1 . A widely used approach for calibrating these parameters underthe existing model (15) follows the method of Piterbarg (2005), which is both efficient and accurate. It turns out thata similar calibration procedure can also be derived for the proposed model. The calibration procedure involves twomain approximation steps: (i) Markovian projection, and (ii) skew averaging. In what follows, we present a detailedderivation of these approximations under the context of the proposed model and subsequently summarize the resultedcalibration procedure.

3.1 Markovian Projection

Consider call options with a maturity T i for which the relevant forward equity price is F (t, T i). Under the proposed

model, the dynamics of F (t, T i) is given by

dF (t, T i)

F (t, T i) = B(t, T i)σr(t)dW

T ir (t) + γ (t, S (t)) dW

T iS (t), (18)

where W T ir (t) and W T iS (t) are two correlated Brownian motions under the forward measure QT i and

γ (t, S ) =

β (t) + (1 − β (t)) x(t)

S (t)

σS (t), (19)

ρdt =⟨

dW T ir (t), dW T iS (t)

⟩. (20)

This step aims to replace γ (t, S (t)) in (18) by

γ̂ (t, T i, F ) = E QT i

γ (t, S (t)) F (t, T i) = F . (21)

Since S (t) = P (t, T i)e

∫ t

0d(u)du

F (t, T i), this boils down to calculating the following conditional expectation

E QT i

x(t)

P (t, T i)

F (t, T i) = F

. (22)

For this purpose, we apply the approximation

γ (t, S (t)) ≈ γ (t, x(t)) = σS (t), (23)to (18), which gives

F (t, T i) ≈ F (0, T i)exp

−12

t

0

λ2(u, T i)du + t

0

B(u, T i)σr(u)dW T ir (u) +

t

0

σS (u)dW T iS (u)

, (24)

where

λ(u, T i) =

B2(u, T i)σ2r(u) + σ

2S (u) + 2ρσS (u)B(u, T i)σr(u)

12 . (25)

Note that with this approximation ln F (t, T i) is normally distributed. On the other hand, we have

x(t)

P (t, T i) = F (0, T i)exp

−V I (0, T i) − V I (t, T i)

2 +

t0

B(u, T i)σr(u)dW T ir (u)

, (26)

which implies ln x(t)P (t,T i)

is also normally distributed. Using (24) and (26), we obtain

E QT

x(t)

P (t, T i)

F (t, T i) = F

≈ F (0, T i)

F

F (0, T i)

χ(t)exp(R(t)) (27)

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where R(t) is a quantity of quadratic order in volatilities and

χ(t) =

∫ t0

B2(u, T i)σ2r(u)du + ρ

∫ t0

σS (u)B(u, T i)σr(u)du

∫ t

0 λ2(u, T i)du

. (28)

Dropping the quadratic term R(t), we have

γ̂ (t, T i, F ) =

β (t) + (1 − β (t))

F (0, T i)

F

1−χ(t)σS (t). (29)

Consequently, we have the following approximated dynamics for F (t, T i):

dF (t, T i)

F (t, T i) = B(t, T i)σr(t)dW

T ir (t) + γ̂ (t, F (t, T i)) dW

T iS (t), (30)

which can be further rewritten in a one–dimensional form

dF (t, T i)

F (t, T i)

= Λ(t, T i, F (t, T i))dW T iF (t), (31)

where W T iF (t) is a standard Brownian motion under QT i and

Λ(t, T i, F ) =

B2(t, T i)σ2r(t) + γ̂

2(t, T i, F ) + 2ργ̂ (t, T i, F )B(t, T i)σr(t) 12 . (32)

3.2 Skew Averaging

Define

g(t, T i, F ) = F Λ(t, T i, F ), (33)

ḡ(T i, F ) = β F (T i) F

F (0, T i) + (1 − β F (T i)). (34)

Following Piterbarg (2003), the forward price dynamics given in (31) can be well approximated by

dF (t, T i) = g(t, T i, F (0, T i))ḡ(T i, F (t, T i))dW T iF (t), (35)

provided that

∂

∂F ḡ(T i, F )

F =F (0,T )

=

T i0

ω(t)

∂ ∂F

g(t, T i, F )F =F (0,T )

g(t, T i, F (0, T i)) dt, (36)

where

ω(t) = u(t)

∫ T i0 u(t)dt

, (37)

u(t) = g2(t, T i, F (0, T i))

t0

g2(s, T i, F (0, T i))ds. (38)

We may write (35) explicitly as

dF (t, T i) = Λ(t, T i, F (0, T i)) [β F (T i)F (t, T i) + (1 − β F (T i))F (0, T i)] dW T iF (t). (39)The condition (36) implies

β F (T i) − 1 =2∫ T i0

(β (t) − 1)∫ t

0 Λ2(u, T i, F (0, T i))du

σ2S (t) + ρσS (t)B(t, T i)σr(t)

χ(t)dt

∫ T i0

Λ2(u, T i, F (0, T i))du2 . (40)

This implies that the approximated forward price follows a DD1F model with skew β F and a time–dependent volatilityparameter Λ(t, T i, F (0, T i)). Notice that the dynamics (39) follows a DD1F model with a constant skew parameter β F

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and time–dependent volatility Λ(t, T i, F (0, T i)). As far as the distribution of F (T i, T i) is concerned, we may replace(39) with the following DD1F dynamics

dF (t, T i) = σF (T i) [β F (T i)F (t, T i) + (1 − β F (T i))F (0, T i)] dW T iF (t), (41)

where

σF (T i) =

√ ∫ T i0

Λ2(u, T i, F (0, T i))du

T i. (42)

Notice that the right hand side of (42) is independent of the model skew parameter β (t).

3.3 The Calibration Procedure

Utilizing the approximations above, the parameters {(β i, σi)}N −1i=0 can be calibrated as follows1. At each maturity T i, where i = 1,...,N , fit a DD1F model with constant skew and volatility parameters β

∗i,F

and σ∗i,F to the observed market implied volatilities, i.e. a model with the following dynamics

dF (t, T i) =

β ∗i,F F (t, T i) + (1 − β ∗i,F )F (0, T i)

σ∗i,F dW (t). (43)

This gives a collection of effective DD1F skew and volatility parameters

(β ∗i,F , σ∗i,F )

N −1i=0

.

2. Solve {σi}N −1i=0 recursively using (42) and replacing σF (T i) by σ∗i,F . (Note that the determination of {σi}N −1i=0does not require the knowledge of {β i}N −1i=0 .)

3. Given {σi}N −1i=0 and

β ∗i,F N −1i=0

, solve {β i}N −1i=0 recursively using (40) and replacing β F (T i) by β ∗i,F .In the first step above, at each maturity, we first set the DD1F volatility parameter equal to the ATM market impliedvolatility and optimize the DD1F skew parameter against the selected market implied volatilities. Next, we adjust theDD1F volatility parameter so that the ATM market implied volatility is matched exactly. This process is fast since it

involves only one–dimensional optimization in which the initial guess is suitably set. The second and third steps of thecalibration procedure are instantaneous since they involve solving a series of quadratic equations and linear equationsonly.

It is worth noting that a similar calibration procedure can also be derived for the existing model based on thedynamics (15). The resulted calibration procedure follows exactly the same steps as above where the only modificationis in the function χ(t). This implies that the calibrated model volatility parameters coincide under the two modelswhile the calibrated skew parameters may differ.

4 Valuation

Derivative pricing under the proposed model can be carried out either by a Monte Carlo simulation method or a finitedifference method. For a general payoff, the pricing partial differential equation (PDE) arising from the proposedmodel is three–dimensional, where the additional factor is attributed to the variable x(t) appearing in the equityvolatility function. The establishment of an efficient and robust implementation of a mesh–based finite differencescheme for a three–dimensional pricing PDE is a highly non–trivial task. Moreover, it is generally difficult to applya finite difference method to strongly path–dependent products such as those with a TARN or snowball feature. Forthese reasons, we choose to devote our efforts to identifying an efficient Monte Carlo simulation method for generalpricing purposes. As we shall see, the strength of the proposed model is indeed best realized within the Monte Carlopricing framework to be developed in this section.

4.1 An Explicit Equity Price Expression

Consider the shifted equity price S̃ (t), defined by

S̃ (t) = S (t) + 1 −β

(t)β (t) x(t), (44)

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Assume k1 ≤ k2 ≤ ·· · ≤ kn. Let Qi,n be a probability measure characterized by the following Radon–NikodymDerivative (RND) process:

dQi,n

dQT i(t) = exp

−n

2 (V I (0, T i) − V I (t, T i))2

+ n

t0

B(u, T i)σr(u)dW T ir (u)

, (54)

where B(·) and V I (·) are defined in (5) and (7), respectively. It is easy to verify that the following holdsdQi,n

dQT i(T i) =

xn(T i)

F n(0, T i) exp

−n(n − 1)

2 V I (0, T i)

. (55)

Applying Girsanov’s Theorem, we conclude that

W i,nr (t) = W T ir (t) − n

t0

B(u, T i)σr(u)du, (56)

W i,nS (t) = W

T iS (t) − nρ

t0

B(u, T i)σr(u)du, (57)

are two correlated Brownian motions under Qi,n with

< dW i,nr (t), dW i,nS (t) > = ρdt. (58)

Consequently, we have

mi,n(k1, k2, · · · , kn) = F n(0, T i)exp

n(n − 1)2

V I (0, T i)

E Qi,n [M (T k1 , T i)M (T k2 , T i) · · · M (T kn, T i)] , (59)

where each M (T kj , T i) can be rewritten as

M (T kj , T i) = exp

−1

2

T iT kj

σ̃2S (u)du + (n − 1)ρ T iT kj

σ̃S (u)B(u, T i)σr(u)du +

T iT kj

σ̃S (u)dW i,nS (u)

. (60)

Since each M (T kj , T i) is lognormally distributed, direct calcualtion gives

mi,n(k1, k2, · · · , kn) = F n(0, T i)expn(n − 1)

2 V I (0, T i) + (n − 1)ρ

n∑j=1

T iT kj

σ̃S (u)B(u, T i)σr(u)du

× expn−1∑j=1

j( j − 1)2

T kj+1T kj

σ̃2S (u)du + n(n − 1)

2

T iT kn

σ̃2S (u)du

. (61)

4.2 An Exact Monte Carlo Pricing Framework

We consider a hybrid instrument consisting of a stream of net payments {Φi}M

i=1 which take place at the times {T p

i }M

i=1,respectively. Each payment Φi takes the form:

Φi = Φi

r

T f i,1

, · · · , r

T f i,ni

, S

T f i,1

, · · · , S

T f i,ni

, (62)

where T f i,1 < T f i,2


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To valuate the instrument above with Monte Carlo simulation under the risk neutral measure Q, the relevantvariables to be simulated are

y

T f 1,1

, · · · , y

T f 1,n1

, S

T f 1,1

, · · · , S

T f 1,n1

,

..

.y

T f M,1

, · · · , y

T f M,nM

, S

T f M,1

, · · · , S

T f M,nM

,

exp

T p10

r(u)du

, · · · , exp

T pM

0

r(u)du

. (63)

From (9), (10), (17), (46), and (48), this reduces to simulating

• the process X (t) = ∫ t0

E (u)σr(u)dW r(u) at

T f i,j

i = 1, · · · , M , j = 1, · · · , ni and {T pi }M i=1,• the process Y (t) =

∫ t

0 B (0, u)E (u)σr(u)dW r(u) at {T pi }M i=1, and

• the process Z (t) = ∫ t0 σ̃S (u)dW S (u) at T f i,j i = 1, · · · , M , j = 1, · · · , ni.To achieve exact simulation via (48), our simulation time grids should include both the parameter time grids {T i}N i=1.For ease of exposition, we simply assume that the time grids {T i}N i=1 consist of all relevant product related time gridsdescribed above.

It is clear that X (t), Y (t), and Z (t) are Gaussian processes and hence can be simulated exactly without the needof path discretization. It suffices to consider the simulation of the following three variables

X =

T iT i−1

E (u)σr(u)dW r(u), Y =

T iT i−1

B(0, u)E (u)σr(u)dW r(u), Z =

T iT i−1

σ̃S (u)dW S (u). (64)

The three variables above are normally distributed with a mean of zero while their variances and covariances can also

be calculated readily. In practice, the variances and covariances can be calculated and cached before carrying out thepricing simulation. Our experience suggests that this can reduce the computational time of pricing considerably. Foreach simulation of the three processes, a sample of the variables in (63) can then be constructed according to (9), (10),(17), (46), and (48) respectively.

It is worth mentioning that the existing model (15) does not lead to any explicit expression for S (t). A pathdiscretization scheme is therefore required for the simulation of the existing model. Hence, Monte Carlo pricing underthe existing model gives rise to both discretization errors and Monte Carlo noises in prices. To reduce the discretizationerrors, one often needs to insert additional simulation time grids which, needless to say, increases the time of the pricingsimulation. In contrast, the exact Monte Carlo simulation scheme presented above requires simulation of the underlyingprocesses only at relevant model and payoff time grids. Consequently, path samples generated from the scheme aresubject to the inherent Monte Carlo noises only and are free of discretization errors. We believe that an exact MonteCarlo simulation scheme not only enhances the efficiency and robustness of pricing but also provides a reliable wayfor price (e.g. PDE or tree prices) and error (e.g. calibration errors) verification.

4.3 Computation of Risk Sensitivities

For hedging purposes, one often needs to compute the sensitivities of an instrument with respect to the underlyingrisk factors. Here, we particularly focus on the calculation of equity deltas and gammas. Suppose that the value of acertain instrument is given by

V (w) = E [g (w)] , (65)

where w is the variable of interest. The first and second order sensitivities of V with respect to w are defined by

∆(w) = ∂V

∂w, Γ(w) =

∂ 2V

∂w2. (66)

These sensitivities often need to be estimated numerically.

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The conventional bump–and–revalue approach attempts to approximate them using finite differences of the pricefunction:

∆(w) ≈ V (w + h) − V (w)h

, Γw ≈ V (w + h) − 2V (w) + V (w − h)h2

, (67)

where h is the perturbation size. Within a finite difference pricing approach, these approximated sensitivities arenaturally obtained in the pricing stage, while within a Monte Carlo pricing approach revaluations of the instrumentare generally required. The bump–and–revalue method has a few potential issues. In particular, it is generally difficultto come up with a robust choice of the perturbation size h under all pricing settings. A good choice of h often dependson, for example, the underlying instrument and the range of model parameters. For a small h or a higher ordersensitivity, the estimate tends to be more vulnerable to numerical noises. This is a particularly the case for MonteCarlo pricing methods due to discretization errors and Monte Carlo noises. To obtain accurate and smooth higherorder sensitivities, one often needs to increase the number of simulations substantially or utilize certain variancereduction techniques to reduce the noises. On the other hand, for a large h, the approximated sensitivities may consistof a significant bias. For a detailed discussion of this approach and the related issues, see Jeckel (2001), for example.

On the other hand, the approach of Glasserman (1996), known as the pathwise method, offers an alternative wayof calculating sensitivities which is not based on taking finite differences of the price function and hence avoids revalu-ations. In contrast to the bump–and–revalue approach, the pathwise method calculates sensitivities by differentiating

the underlying payoff function with respect to the variables of interest and is naturally suited for a Monte Carlo pricingframework. With the pathwise method, the sensitivities above are given by

∆(w) = E

∂g (w)

∂w

, Γ(w) = E

∂ 2g (w)

∂w2

. (68)

Consequently, one may obtain the sensitivities by evaluating the expected values above through Monte Carlo sim-ulation. Since the calculation of sensitivities can be carried out within the same simulation loop for pricing, thismethod is generally more efficient than the bump–and–revalue method. In essence, the pathwise method involvesan interchange of the order of differentiation and expectation, which, technically speaking, imposes some smoothnessconditions on the payoff function. A detailed discussion of such conditions can be found in Glasserman (1996). InGlasserman (2003), it is reported that the pathwise method, when applicable, tends to produce more accurate greekestimates with lower variances.

We now show that under the current model it is extremely simple to calculate equity deltas and gammas using thepathwise method along with our exact Monte Carlo simulation scheme. To fix ideas, consider the payment Φi of theinstrument defined in Section 4.2. The value of the payment and the pathwise delta and gamma are given by

V i(S 0) = E

Φi

r

T f i,1

, · · · , r

T f i,ni

, S

T f i,1

, · · · , S

T f i,ni

, (69)

∆i(S 0) = E

ni∑j=1

∂ Φi

∂S (T f i,j)

∂S (T f i,j)

∂S 0

, (70)

Γi(S 0) = E

ni∑j=1

∂ 2Φi

∂S 2(T f i,j)

∂S (T f i,j)

∂S 0

2+

∂ Φi

∂S (T f i,j)

∂ 2S (T f i,j)

∂S 20

. (71)

Note that the partial derivatives of the payoff with respect to the underlying equity prices depend only on the form of the payoff, while the model–dependent part lies in the evaluation of the partial derivatives

∂S (T f i,j)

∂S 0,

∂ 2S (T f i,j)

∂S 20. (72)

By applying (49), we obtain

∂ nS (T )

∂S n0=

S (T )S 0

, n = 1,

0, n > 1.

(73)

For payoffs involving a running maximum and/or minimum of the equity prices sampled at a series of time points,one may also need to calculate the following partial derivatives

∂ nM (T 1, · · · , T i; η)∂S n0

, (74)

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where

M (T 1, · · · , T i; η) = η max {ηS (T 1), · · · , ηS (T i)} , η = ±1. (75)Applying (49) and using the following property

M (T 1, · · · , T i; η) = ηS 0 maxη S (T 1)S 0

, · · · , η S (T 1)S 0

. (76)

we also obtain

∂ nM (T 1, · · · , T i; η)∂S n0

=

M (T 1,···,T i;η)S 0

, n = 1,

0, n > 1.

(77)

5 Numerical Illustration

As an illustration, we apply the proposed model and the results developed earlier to evaluate a conditional trigger swap(CTS), also known as an equity Enmansai in Japanese markets. An Enmansai is essentially a swap contract involving

the exchange of cash flows between two counterparties over time, with the funding leg paying Libor–plus–spreadcoupons and the structured leg paying equity–linked structured coupons. In addition, the contract is auto–callablewith an up–and–out (UO) condition. Consequently, the contract terminates either on the maturity of the trade orthe first monitoring time at which the equity price exceeds the UO barrier, whichever comes first. In essence, CTSsare designed to pay high coupons as long as the underlying equity index (typically the Nikkei 225) is above a certainlevel, and redeem out if it further climbs above a pre–defined barrier. This explains the great success of these productsduring the Japanese equity market rally (2003–2007). Although the demand for CTSs diminishes substantially dueto the recent market deterioration, many banks still have large blocks of active positions in CTSs. For these banks, itremains an important task to evaluate and risk–manage their CTS positions appropriately.

In practice, there exists variations of CTS structures. Here, we consider only a basic form of the product, asdescribed in Overhaus et al (2007). With this structure, the structured leg pays an initial guaranteed coupon C H (adjusted by an accrual factor τ 1) at T 1 and subsequently a structured coupon in the form

τ i

C H I {S (T i)≥K } + C LI {S (T i)


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• market implied volatilities: see Table 4.• time grids: {T i = 0.5 · i}60i=1• HW1F model parameters: a(t) = 0.75% and σr(t) = 0.7%.

• model correlation: ρ = 0.6.The data and parameter settings above are chosen so that they roughly reflect the recent market conditions.

5.2 Test Results

5.2.1 Calibration

We apply the calibration procedure described in Section 3 along with the market data and parameter settings describedabove to determine the model parameters {(σi, β i)}N −1i=0 . With this calibration method, the calibration time is typicallywithin a few seconds. Fig. 2 and Fig. 3 show both the fitted DD1F parameters and model parameters.

Table 5 shows the differences between the model implied volatilities and the actual market implied volatilities.The fit of the model at most calibration points is excellent with fitting errors typically less than one volatility point.

There remains a few individual points with large fitting errors, which are mostly associated with a short maturity(6M/1Y) and a deep OTM/ITM moneyness. Since short maturity ITM/OTM options usually have a small vega risk,we believe that these poorly fitted points should not give rise to significant pricing impacts on long dated CTS trades.It also appears that the quality of the fit of the model at longer maturities deteriorates slightly. However, the fittederrors in such cases are still well within bid ask spreads. We believe that the overall calibration results achieved inthis example are sufficiently accurate for practical pricing purposes. More importantly, the calibrated model gives anadequate accounting of the term structure of implied volatility skews exhibited in the market data.

It should be pointed out that the poorly fitted cases are largely due to a limitation of the underlying model ratherthan the calibration algorithm. To see this, we show in Table 6 the differences between the model implied volatilitiesand the fitted DD1F implied volatilities. The fit of the model to the fitted DD1F implied volatilities is excellent. Inparticular, all fitted errors for maturities up to 15Y are below 0.50% while for maturities beyond 15Y there exists onlyone case with a fitted error above one volatility point. Consequently, the fitting capability of the underlying modelis to some extent reflected in the fitting capability of the DD1F models to market option prices. This implies that

the underlying model may not be well fitted to convex implied volatility smiles or skews beyond the limits of a DD1Fmodel.

5.2.2 Pricing

We evaluate two test CTS trades described in Table 1 using the exact simulation scheme presented in Section 4 alongwith the calibration results obtained in the previous section. For comparison, the trades are also evaluated using threecommon path–discretization based simulation schemes (Euler, Milstein, and predictor–corrector (PC)), respectively.Tables 7 and 8 report the relative price errors under different choices of simulation sizes and step sizes, where thebenchmark prices are obtained using the exact scheme with a simulation size of 10 millions. It should be emphasizedthat a fair comparison of the four simulation schemes should be based on both price accuracy and computationalefficiency. For this reason, we show in Table 9 the computational times taken for the pricing of Trade 2. The pricingtimes for Trade 1 are similar and hence are omitted.

Table 1: Contract parameters of two test trades.

- notional maturity coupon frequency payment accrual C L C H K U sTrade 1 1 30Y semi–annual 0.5 0.1% 2% 12000 15000 0.0Trade 2 1 30Y semi–annual 0.5 0.1% 2% 10000 12000 0.0

For Trade 1, the relative price errors under the exact simulation scheme are within 0 .50% for all selected simulationsizes. Under the other three schemes, it appears that a step size of 0.05 or smaller is required to obtain reliable prices.With a step size of 0.05 and a simulation size of 50K or more, the relative price errors under the three schemes arewithin 0.50%. For this trade, the Milstein scheme does not show a clear advantage over the Euler scheme in price

convergence, while the overall performance of the PC scheme appears to be better than the other two schemes. As abenchmark comparison for the current trade, using a simulation size of 50K and a step size of 0.05 (under the three

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discretization based schemes), the computational times required for the four simulation schemes are 1.67 seconds, 9.97seconds, 10.28 seconds, and 10.92 seconds, respectively, as shown in Table 9.

On the other hand, the relative price errors for Trade 2 appear to be larger in general under all simulation schemes.The price convergence under the exact scheme remains relatively well behaved. In particular, the results indicate thata relative price error of 0.50% or below can be achieved with a simulation size between 100K and 200K. In contrast,

the other three schemes show slower price convergence for most step sizes. It appears that a step size of 0.01 and asimulation size of 50K or more are required to obtain reliable prices for Trade 2 under the three schemes. For thistrade, both the Milstein scheme and the PC scheme show a clear improvement in price convergence over the Eulerscheme, particularly for larger step sizes. As a benchmark comparison, using a simulation size of 100K under the exactscheme and a simulation size of 50K together with a step size of 0.01 under the three discretization based schemes, thecomputational times required for the four schemes are 3.42 seconds, 49.22 seconds, 51.11 seconds, and 54.62 seconds,respectively, as shown in Table 9.

It is worth noting that for Trade 2 the price convergence under the three discretization based simulation schemesis more sensitive to the step size compared to the case of Trade 1. In particular, a smaller step size appears to berequired for a reasonable price convergence for Trade 2. This is indeed expected. Since the coupon strike of Trade2 coincides with the spot price and the UO barrier is also closer to the spot price, any numerical errors can moreeasily translate into large impacts on the coupon payoffs due to the payoff discontinuities at the coupon strike andUO barrier. This, on one hand, explains why the price errors under all four simulation schemes are generally largerin the case of Trade 2. On the other hand, it also implies that the price errors under the three discretization basedsimulation schemes become more sensitive to the step size since a larger step size results in larger discretization errors.In contrast, the price errors under the four simulation schemes appear to be less sensitive to Monte Carlo noises anddiscretization errors in the case of Trade 1. This is a consequence of two effects: (i) both the coupon strike and theUO barrier of Trade 1 are relatively far from the spot price, and (ii) both C H and C L are independent of the equityprices. In practice, there exist CTS trades where the structured coupons are a series of call options on the underlyingequity prices. For these trades, each coupon payoff depend directly on the relevant equity price in that each level of the equity price leads to a different coupon payff. Consequently, the choice of the step size under the discretizationbased simulation schemes can become more critical in order to obtain sufficiently accurate price figures.

Our numerical experience also suggests that the discretization based schemes perform worse when the volatility ishigh. This is expected since path discretization involves freezing the drift and the volatility of the underlying processin a certain manner over each discretization step and hence ignores some stochasticity of the process increments. As

a result, a high volatility can lead to larger distributional bias and hence larger price bias. The overall test resultssuggest that the exact simulation scheme presented in this article is superior to the discretization based simulationschemes taking into account both price accuracy and computational efficiency. It is also more robust in that it avoidsthe ambiguity in choosing a step size which is otherwise needed for the discretization based simulation schemes. As isevident from the test results, choosing an appropriate step size that can strike a good balance between computationalefficiency and price accuracy is not always trivial. In particular, it can depend on the type of CTS coupons, thediscretization scheme, and the levels of the contract and model parameters.

5.2.3 Equity Delta and Gamma

Let ∆i(S 0) and ∆fundingi (S 0) denote the equity deltas of the structured leg coupon and the funding leg coupon paid

at T i, respectively. Applying the pathwise method described in Section 4.3, we have

∆i(S 0) = τ i E −M i

S 0δ U (M i)Θi + S (T

i)S 0

I {M i


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To evaluate the equity deltas by simulation, one needs to approximate δ U (·) and δ K (·) by some ordinary function. Asimple choice is to approximate them by a normal density function, i.e.

δ̃ b(w, ν ) = 1√

2πν e−

(w−b)2

2ν2 , (86)

where ν is a tuning parameter which, roughly speaking, controls how close the approximating function is to the trueDirac delta function. Consequently, we have

∆i(S 0) ≈ τ i E −M i

S 0δ̃ U (M i, ν M )Θi +

S (T i)

S 0I {M i


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skew parameters to 1 while the volatility parameters are chosen to reproduce the market ATM implied volatilities.The results clearly indicate that the prices of the two trades are extremely sensitive to implied volatility skews. Inparticular, taking into account the implied volatility skews, the prices of the two trades are adjusted down by 21.28%and 40.60%, respectively.

Table 2: Impacts of implied volatility skews.

Trade 1 Trade 2Model structured leg funding leg net structured leg funding leg net

hybrid Black model 0.04877 0.18366 0.13488 0.04081 0.09872 0.05791proposed model 0.05624 0.16241 0.10617 0.04249 0.07689 0.03440

relative diff. 15.30% -11.57% -21.28% 4.11% -22.12% -40.60%

* The tests utilize the exact simulation scheme with a simulation size of 100K.

5.2.5 Impacts of Stochastic Interest Rates

In Fig. 4 and Fig. 5, we plot the values of Trade 1 and Trade 2 as a function of the equity and interest ratecorrelation parameter ρ. For comparison, we also display the values of the trades under deterministic interest rateswhere we simply set the HW1F volatility to zero and calibrate the model accordingly. It is clear that the impacts of the correlation on the funding leg are much more significant than those on the structured leg. For both trades, thefunding leg value and also the trade value are decreasing in the correlation while the structured leg values are slightlyincreasing in the correlation. These observations are consistent with those shown in Overhaus (2007).

5.2.6 Comparison to the Existing Model

Fig. 22 and Fig. 23 depict the calibrated skew and volatility parameters under the proposed model and the existinghybrid model based on the dynamics (15), respectively. Notice that the calibrated volatility parameters {σi}N −1i=0coincide under the two models, which is evident from the calibration procedure described in Section 3 On the otherhand, the skew parameters under the proposed model appear to be lower than those under the existing model.

As a further comparison, we show in Fig. 24 – Fig. 35 the coupon values, equity deltas, and equity gammas of the two test trades obtained under the two models. Since only discretization based simulation schemes can be appliedunder the existing model, the tests utilize a PC simulation scheme under the existing model and the exact simulationscheme under the proposed model. The simulation size is fixed to 100K under both models while a step size of 0.01is used under the existing model. Our numerical experience suggests that these settings tend to give reliable pricingresults under typical market conditions and contract parameters. On the other hand, the equity deltas and gammasare calculated using a pathwise method. From the figures, it is clear that both models lead to similar prices, deltas,and gammas. Hence, computational efficiency in pricing is a natural criterion for choosing between the existing modeland the proposed model.

To this end, we report in Table 3 the computational times taken for the pricing of Trade 2 under the two models.The simulation size and step size are chosen for a benchmark comparison. As shown in the table, the proposed model,owing to the existence of an exact Monte Carlo pricing scheme, is much more efficient than the existing model in

pricing. In particular, it requires less than 4 seconds for calculating price only (about 30 times faster than the existingmodel) and less than 5 seconds when including equity deltas and gammas (about 20 times faster than the existingmodel). To achieve a comparable performance, for example, one would have to decrease the simulation size to near20K and increase the step size to 0.05 under the existing model. From our previous test results, such settings arenot reliable in general under the existing model. In addition, as is evident from our previous discussion, the proposedmodel is also more robust than the existing model since it avoids the ambiguity in choosing a step size. The overallresults suggest that the proposed model is superior to the existing model from the viewpoints of both computationalefficiency and robustness in pricing.

6 Conclusion

This article proposes a parsimonious pricing model for CTSs which can capture the effects of implied volatility skews

and stochastic interest rates simultaneously. A fast and accurate calibration procedure along the lines of Piterbarg(2005) is presented. The proposed model is shown to be an improvement of an existing hybrid model that is oftenused by market practitioners for the pricing of CTSs. In particular, it leads to an explicit equity price expression and

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Table 3: Computational time for Trade 2 (in seconds).

computational time* proposed model** existing model***price only 3.25 97.68

price, equity deltas, and equity gammas 4.75 102

* The tests are run on a 3.16GHz Duo CPU with 3.50 GB RAM.** The exact simulation scheme is used with a simulation size of 100K.*** The PC simulation scheme is used with a simulation size of 100K and a step size of 0.01.

consequently an exact Monte Carlo pricing scheme. Numerical results indicate that the exact simulation scheme ismore efficient and robust than discretization based simulation schemes. We also confirm that the effects of impliedvolatility skews and stochastic interest rates can be very significant and cannot be neglected in the pricing of CTSs.A further comparison between the proposed model and the existing model reveals that both models lead to similarprices, equity deltas, and equity gammas for CTSs. However, with the exact simulation scheme, the proposed modelis shown to be a more appealing alternative to the existing model due to its superior computational efficiency androbustiness in pricing.

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References

[1] Brigo D. and Mercurio F., Interest Rate Models: Theory and Practice, 2nd ed., Springer , 2001.

[2] Broadie M. and Glasserman P., Estimating Security Price Derivatives Using Simulation, Management Science ,1996, V42:2.

[3] Glasserman P., Monte Carlo Methods in Financial Engineering, 1st ed., Springer , 2003.

[4] Gurrieri S., Nakabayashi M., and Wong T., Calibration of the Hull–White Short Rate Model, working paper , 2009.

[5] Hagan P., Evaluating and Hedging Exotic Swap Instruments via LGM, Bloomberg Technical Report..

[6] Jeckel P., Monte Carlo Methods in Finance, 1st ed., John Wiley & Sons , 2001.

[7] Longstaff A. and Schwartz S., Valuing American Options by Simulation, Review of Financial Studies , 2001, V14:113–148.

[8] Marris D., Financial Option Pricing and Skewed Volatility, MPhil Thesis, University of Cambridge , 1999.

[9] Muck M., On the Similarity between Displaced Diffusion and Constant Elasticity of Variance Market Models of the Term Structure, WHU , 2005.

[10] Overhaus M., Bermudez A, Buehler H., Ferraris A., Jordinson C., and Lamnouar A., Equity Hybrid Derivatives,John Wiley & Sons , 2007.

[11] Piterbarg V., A Multi–Currency Model with FX volatility skew, SSRN Working Paper , 2005.

[12] Rebonato R., Volatility and Correlation: The Perfect Hedger and the Fox, 2nd ed., John Wiley & Sons , 2004.

[13] Svoboda-Greenwood S., The Displaced Diffusion as an Approximation of the CEV, Appl. Math. Finance , 2009,V66: 269–286.

[14] Wong T., An Analytical Hybrid Equity and Interest Rate Model, SSRN working paper , 2009.

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APPENDIX

A Tables and Figures

Figure 1: JPY zero rates.

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Figure 2: Calibrated model volatilities.

Figure 3: Calibrated model skews.

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Table 4: Market implied volatilities of equity options at some selected moneyness (quoted as % of forward prices) andmaturities.

Maturity/Moneyness 0.4 0.6 0.8 1.0 (ATM) 1.2 1.4 1.66M 39.00% 35.90% 26.99% 20.26% 16.98% 15.84% 15.88%1Y 38.15% 33.39% 26.11% 20.55% 17.05% 14.91% 13.61%2Y 37.14% 31.28% 25.51% 21.13% 18.05% 15.84% 14.20%3Y 35.58% 29.98% 25.24% 21.68% 19.09% 17.15% 15.65%4Y 34.28% 29.20% 25.19% 22.21% 20.02% 18.35% 17.05%5Y 33.30% 28.72% 25.26% 22.72% 20.84% 19.40% 18.26%6Y 32.54% 28.43% 25.41% 23.21% 21.57% 20.31% 19.32%7Y 32.03% 28.31% 25.63% 23.67% 22.22% 21.10% 20.21%8Y 31.66% 28.27% 25.86% 24.12% 22.82% 21.81% 21.01%9Y 31.41% 28.30% 26.12% 24.54% 23.37% 22.46% 21.73%

10Y 31.27% 28.40% 2 6.40% 24.95% 23.87% 23.03% 22.36%15Y 31.69% 29.42% 2 7.84% 26.69% 25.81% 25.12% 24.56%20Y 32.24% 30.31% 2 8.96% 27.96% 27.20% 26.59% 26.10%30Y 32.68% 31.13% 3 0.05% 29.24% 28.61% 28.11% 27.69%

Table 5: Calibration errors in implied volatilities: model vs market, number of simulations = 1 million.

Maturity/Moneyness 0.4 0.6 0.8 1.0 (ATM) 1.2 1.4 1.66M 3.39% -3.51% -1.45% 0.00% -1.13% -3.93% -7.91%1Y 0.55% -3.07% -1.40% -0.01% 0.19% -0.40% -1.37%2Y 0.65% -1.26% -0.62% -0.01% 0.11% -0.10% -0.50%3Y 0.78% -0.58% -0.33% -0.01% 0.09% 0.06% -0.06%4Y 0.84% -0.29% -0.21% 0.00% 0.10% 0.13% 0.12%5Y 0.85% -0.12% -0.12% 0.02% 0.12% 0.17% 0.20%6Y 0.86% -0.02% -0.06% 0.05% 0.14% 0.21% 0.25%7Y 0.87% 0.05% -0.01% 0.08% 0.17% 0.24% 0.29%8Y 0.87% 0.11% 0.04% 0.11% 0.19% 0.26% 0.31%9Y 0.86% 0.16% 0.09% 0.15% 0.22% 0.28% 0.33%

10Y 0.86% 0.21% 0.13% 0.18% 0.24% 0.30% 0.35%15Y 0.94% 0.36% 0.27% 0.30% 0.37% 0.43% 0.49%20Y 1.09% 0.52% 0.42% 0.44% 0.50% 0.56% 0.62%30Y 1.51% 0.90% 0.75% 0.74% 0.77% 0.82% 0.87%

Table 6: Differences in implied volatilities: model vs fitted DD1F, number of simulations = 1 million.

Maturity/Moneyness 0.4 0.6 0.8 1.0 (ATM) 1.2 1.4 1.66M 0.06% 0.01% 0.00% 0.00% 0.00% 0.05% 0.20%1Y 0.25% 0.09% 0.01% -0.01% 0.02% 0.11% 0.33%2Y 0.12% 0.04% 0.00% -0.01% 0.00% 0.05% 0.12%

3Y 0.19% 0.06% 0.00% -0.01% 0.00% 0.05% 0.13%4Y 0.25% 0.09% 0.02% 0.00% 0.02% 0.06% 0.11%5Y 0.30% 0.12% 0.04% 0.02% 0.03% 0.06% 0.10%6Y 0.33% 0.15% 0.07% 0.05% 0.05% 0.07% 0.10%7Y 0.35% 0.18% 0.11% 0.08% 0.08% 0.09% 0.10%8Y 0.38% 0.21% 0.14% 0.11% 0.10% 0.10% 0.11%9Y 0.40% 0.24% 0.18% 0.15% 0.13% 0.12% 0.12%

10Y 0.41% 0.28% 0.21% 0.18% 0.16% 0.14% 0.13%15Y 0.47% 0.40% 0.35% 0.30% 0.27% 0.24% 0.21%20Y 0.61% 0.55% 0.49% 0.44% 0.40% 0.35% 0.31%30Y 1.07% 0.92% 0.82% 0.74% 0.67% 0.61% 0.56%

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T a b l e 7 : R e l a t i v e p r i c e

e r r o r s o f T r a d e 1 : t r u e p r i c e = 0 .

1 0 6 3 5 8 ( b a s e d o n t h e e x a c t s c h e m

e w i t h 1 0 m i l l i o n s i m u l a t i o n s ) .

E u l e r s c h e m e

M i l s t e i n s c h e m e

P C

s c h e m e

n u m . o f s i m u l a t i o n

E x a c t

s t e p =

0 . 5

s t e

p =

0 . 1

s t e p =

0 . 0 5

s t e p =

0 . 0 1

s t e p =

0 . 5

s t e p =

0 . 1

s t e p =

0 . 0 5

s t e p =

0 . 0 1

s t e p =

0 . 5

s t e p =

0 . 1

s t e

p =

0 . 0 5

s t e p =

0 . 0 1

1 0

0 . 4 1 %

1 . 7 7 %

0

. 5 6 %

0 . 2 6 %

- 0 . 1 1 %

2 . 8 2 %

0 . 9 4 %

0 . 2 5 %

- 0 . 0 3 %

0 . 9 6 %

0 . 4 2 %

0 . 0 6 %

- 0 . 0 8 %

2 0

0 . 0 9 %

1 . 9 1 %

1

. 2 2 %

1 . 1 2 %

0 . 3 9 %

2 . 8 6 %

1 . 5 5 %

1 . 2 8 %

0 . 4 4 %

1 . 0 0 %

1 . 0 9 %

1 . 0 7 %

0 . 3 9 %

3 0

0 . 3 0 %

2 . 2 1 %

1

. 1 5 %

0 . 8 7 %

0 . 2 9 %

2 . 9 5 %

1 . 3 2 %

1 . 0 5 %

0 . 3 1 %

0 . 9 8 %

0 . 8 8 %

0 . 8 5 %

0 . 2 8 %

5 0

- 0 . 5 0 %

1 . 3 6 %

0

. 4 4 %

0 . 1 9 %

- 0 . 1 4 %

2 . 2 2 %

0 . 8 3 %

0 . 3 4 %

- 0 . 1 3 %

0 . 2 8 %

0 . 3 6 %

0 . 1 6 %

- 0 . 1 6 %

1 0 0

- 0 . 1 8 %

1 . 5 8 %

0

. 7 6 %

0 . 5 3 %

0 . 2 9 %

2 . 4 0 %

1 . 0 6 %

0 . 6 3 %

0 . 2 8 %

0 . 4 1 %

0 . 6 5 %

0 . 4 6 %

0 . 2 5 %

2 0 0

- 0 . 2 5 %

1 . 4 9 %

0

. 5 8 %

0 . 3 3 %

0 . 1 6 %

2 . 3 1 %

0 . 8 3 %

0 . 5 1 %

0 . 1 5 %

0 . 3 1 %

0 . 4 1 %

0 . 3 2 %

0 . 1 2 %

5 0 0

- 0 . 1 7 %

1 . 4 2 %

0

. 3 4 %

0 . 1 9 %

0 . 0 0 %

2 . 2 3 %

0 . 5 9 %

0 . 3 0 %

0 . 0 2 %

0 . 2 1 %

0 . 1 8 %

0 . 1 1 %

- 0 . 0 2 %

T a b l e 8 : R e l a t i v e p r i c e

e r r o r s o f T r a d e 2 : t r u e p r i c e = 0 .

0 3 4 6 2 6 ( b a s e d o n t h e e x a c t s c h e m

e w i t h 1 0 m i l l i o n s i m u l a t i o n s ) .



P C

s c h e m e


E x a c t

s t e p =

0 . 5

s t e

p =

0 . 1

s t e p =

0 . 0 5

s t e p =

0 . 0 1

s t e p =

0 . 5

s t e p =

0 . 1

s t e p =

0 . 0 5

s t e p =

0 . 0 1

s t e p =

0 . 5

s t e p =

0 . 1

s t e

p =

0 . 0 5

s t e p =

0 . 0 1

1 0

- 4 . 1 6 %

6 . 1 3 %

- 0 . 7 3 %

- 1 . 3 2 %

- 2 . 2 6 %

1 . 0 7 %

- 0 . 3 5 %

- 3 . 4 6 %

- 2 . 4 8 %

- 2 . 0 1 %

- 1 . 6 8 %

- 3 . 7 6 %

- 2 . 5 2 %

2 0

- 2 . 5 2 %

9 . 6 9 %

4

. 3 0 %

4 . 4 4 %

2 . 4 1 %

5 . 5 8 %

4 . 0 3 %

2 . 9 6 %

2 . 4 0 %

2 . 0 5 %

3 . 1 4 %

2 . 7 2 %

2 . 3 0 %

3 0

- 1 . 6 0 %

9 . 3 0 %

3

. 8 1 %

3 . 3 6 %

2 . 0 7 %

4 . 6 5 %

3 . 3 0 %

2 . 2 4 %

2 . 2 1 %

1 . 1 6 %

2 . 4 6 %

1 . 9 5 %

2 . 1 3 %

5 0

- 2 . 1 7 %

6 . 7 7 %

2

. 1 7 %

1 . 4 5 %

0 . 2 0 %

3 . 1 3 %

1 . 5 2 %

0 . 6 9 %

0 . 2 7 %

- 0 . 3 4 %

0 . 7 8 %

0 . 2 3 %

0 . 2 1 %

1 0 0

- 0 . 6 5 %

7 . 0 3 %

2

. 4 7 %

2 . 2 7 %

0 . 6 8 %

4 . 2 2 %

1 . 8 0 %

1 . 7 5 %

0 . 9 4 %

0 . 4 4 %

0 . 9 7 %

1 . 3 4 %

0 . 8 3 %

2 0 0

- 0 . 2 7 %

7 . 2 0 %

2

. 1 9 %

1 . 8 8 %

0 . 6 6 %

4 . 4 1 %

1 . 7 3 %

1 . 3 1 %

0 . 8 5 %

0 . 6 3 %

0 . 9 9 %

0 . 9 2 %

0 . 7 5 %

5 0 0

0 . 1 4 %

6 . 7 2 %

1

. 5 0 %

0 . 9 2 %

0 . 2 9 %

3 . 9 5 %

0 . 9 2 %

0 . 3 6 %

0 . 3 7 %

0 . 2 4 %

0 . 2 3 %

- 0 . 0 1 %

0 . 2 9 %

T a b l e 9 : C o m p u t a t i o n a l t i m e s

f o r t h e p r i c i n g o f T r a d e 2 ( i n s e c o n d s ) .



P C s c h e m

e


E x a c t

s t e p =

0 . 5

s t e p

=

0 . 1

s t e p =

0 . 0 5

s t e p =

0 . 0 1

s t e p =

0 . 5

s t e p =

0 . 1

s t e p =

0 . 0 5

s t e p =

0 . 0 1

s t e p =

0 . 5

s t e p =

0 . 1

s t e p

=

0 . 0 5

s t e p =

0 . 0 1

1 0

0 . 3 4

0 . 2 7

1

. 0 3

2 . 0 0

9 . 9 8

0 . 2 7

1 . 0 6

2 . 0 6

1 0 . 5 8

0 . 2 8

1 . 1 3

2 . 1 9

1 0 . 8 4

2 0

0 . 6 7

0 . 5 2

2

. 0 6

3 . 9 8

2 0 . 1 2

0 . 5 2

2 . 1 3

4 . 1 3

2 1 . 1 6

0 . 5 5

2 . 2 7

4 . 3 8

2 2 . 0 6

3 0

1 . 0 2

0 . 7 7

3

. 0 9

5 . 9 8

2 9 . 8 9

0 . 7 8

3 . 1 9

6 . 2 0

3 1 . 4 2

0 . 8 3

3 . 3 8

6 . 5 6

3 3 . 1 2

5 0

1 . 6 7

1 . 2 8

5

. 1 6

9 . 9 7

4 9 . 2 2

1 . 3 1

5 . 3 6

1 0 . 2 8

5 1 . 1 1

1 . 3 8

5 . 6 4

1 0 . 9 2

5 4 . 6 2

1 0 0

3 . 4 2

2 . 5 3

1

0 . 3 0

1 9 . 9 2

1 0 0 . 0 3

2 . 6 1

1 0 . 6 7

2 0 . 6 1

1

0 3 . 1 4

2 . 7 5

1 1 . 3 3

2 1 . 8 6

1 0 8 . 0 9

2 0 0

6 . 6 4

5 . 0 8

2

0 . 6 4

3 9 . 8 9

1 9 7 . 0 6

5 . 2 7

2 1 . 3 3

4 1 . 1 6

2

0 4 . 1 8

5 . 5 0

2 2 . 5 6

4 3 . 6 4

2 1 1 . 9 5

5 0 0

1 6 . 6 4

1 2 . 7 0

5

1 . 6 4

1 0 1 . 9 7

5 0 3 . 9 6

1 3 . 1 4

5 3 . 7 5

1 0 3 . 3 7

5

1 0 . 0 7

1 3 . 7 0

5 6 . 4 7

1

0 9 . 5 6

5 4 7 . 2 0

1 0 0 0

3 3 . 5 3

2 5 . 3 6

1 0

3 . 4 7

1 9 9 . 5 6

9 8 5 . 4 3

2 6 . 3 6

1 0 6 . 6 4

2 0 6 . 1 8

1 0 2 9 . 1 4

2 7 . 3 8

1 1 2 . 9 7

2

1 9 . 0 7

1 0 6 1 . 2 9

* T h e t e s t s a r e r u n o n a

3 . 1 6 G H z D u o C P U

w i t h 3 . 5 0 G B R A M .

21


22/37

Figure 4: Values of Trade 1 as a function of the model correlation. (DIR refers to deterministic interest rate.)

Figure 5: Values of Trade 2 as a function of the model correlation. (DIR refers to deterministic interest rate.)

22


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24/37

Figure 8: Equity deltas of Trade 2.

Figure 9: Equity deltas of Trade 2.

24


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Figure 10: Equity gammas of Trade 2.


25


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26


27/37



27


28/37



28


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29


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30


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Figure 22: Calibrated skew parameters under the proposed model and the existing model.

Figure 23: Calibrated volatility parameters under the proposed model and the existing model.

31


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Figure 24: Structured leg coupon values of Trade 1: existing model vs proposed model.

Figure 25: Funding leg coupon values of Trade 1: existing model vs proposed model.

32


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Figure 26: Equity deltas of funding leg coupons of Trade 1: existing model vs proposed model.


33


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Figure 28: Equity gammas of structured leg coupons of Trade 1: existing model vs proposed model.


34


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Figure 30: Structured leg coupon values of Trade 2: existing model vs proposed model.

Figure 31: Funding leg coupon values of Trade 2: existing model vs proposed model.

35


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36


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Mizuho Enmansai

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